Periodic Solutions And Complicated Dynamics In The State-dependent Delayed Homogeneous And Heterogeneous AIMD/RED Network Congestion Systems

The congestion control system based on Additive Increase and Multiplicative Decrease(AIMD)congestion control and Random Early Detection(RED)algorithm plays a key role in the overwhelming success of the Internet.However,in the previous studies,the roundtrip delay is usually taken as a constant,which is unrealistic.So under the support of the Natural National Science Foundation of China(Nos.11972327 and 11372282),the periodic oscillations and complicated dynamics of the state-dependent round-trip delayed AIMD/RED congestion control system with homogeneous flow and heterogeneous flows,and their modified version are investigated detailedly in this dissertation.The round-trip delay R(t)is not a constant,but depends on the average queue length q(t)of the network congestion control system.So it is a state-dependent delay.Due to the weakening of the smoothness of the righthand of the equation,it is very difficult to study the dynamics of state-dependent delay system.Also its theory is very immature.Firstly,employing the semi-analytical and semi-numerical method called as the harmonic balance method with alternating frequency/time(HB-AFT)domain technique,the approximate analytical expressions of periodic solutions of the generalized homogeneous-flow and heterogeneous flows AIMD/RED systems with state-dependent round-trip delay are considered.Comparing them with the results of numerical simulations by WinPP,they agree very well with each other.It demonstrates that the method employed here is versatile,valid and simple.Then to the end of improving the performance of the systems,we improve the above systems by constructing the packet loss probability function with a hyperbolic tangent function,and obtain the approximate analytical expressions of periodic solutions accurately.In addition,we find that the dynamical phenomena of the improved delayed homogeneous-flow and heterogeneous flows AIMD/RED congestion control systems with state-dependent round-trip delay are different.For the delayed homogeneous system,we find four kinds of bi-stability,i.e.,the coexistence of chaos and Period-3 solution,that of Period-1 and Period-2 solutions,that of Period-2 and Period-2 solutions,that of Period-4 and Period-2 solutions.And the route to chaos,i.e.,Period Doubling bifurcation to chaos,and the window of Period-3 to chaos.For the delayed heterogeneous system,we find that the Period-1 to torus,Period-1 to Period-2 to Period-3 solutions and two kinds of mechanisms of chaos,i.e.,the window of Period-2,the window of Period-3 to chaos.At the same time,it shows that the interaction between the packet loss probability function and the state-dependent delay in the improved systems leads to the occurrence of complex dynamical phenomena in the network congestion control systems.The periodic oscillation can reduce the link utilization,induce the TCP stream synchronization services and further congestion.Chaotic oscillation may result in collapse.Therefore,all complex dynamical phenomena found in this paper are harmful and should be avoided.The obtained results can be very helpful for the researchers to have a better understanding of the mechanism of the network congestion control system,and they can select the parameters properly to improve the network stability and performance.The main features and innovations of this dissertation are as follows.Firstly,the generalized homogeneous/heterogeneous flows AIMD/RED system with state-dependent round-trip delay are mainly studied in this dissertation,which is more practical than the studies on the case of constant round-trip delay in the previous literatures.Secondly,the approximate expressions of the periodic solutions of the systems with state-dependent round-trip delay are obtained accurately by HB-AFT method.Last but not least,in order to make the original systems more reasonable,the packet loss probability function is improved,then the improved AIMD/RED network congestion systems are studied detailedly.The bi-stability and chaos of the improved systems are found,where the window of Period-2 to chaos is rare.